In this appendix we make a brief digression about the methods developed in [1,2]. The main idea is the following: in a system with fast varying coefficients it is possible to construct a change of variables which reveals a shift in the coefficients of the averaged system, which may result in a displacement of the bifurcation value. Such a phenomenon may be responsible of gain or loss of stability of the equilibria, see [1,2]. In [13], we have already discussed the case of a Van der Pol oscillator with rapidly varying coefficients, using the methods developed in [1,2]. In fact the discussion can be generalized so to embrace the general case of an AH bifurcation pattern. Again we just give the main ideas remanding the interested reader to [13, §3.4] for details, see also [2]. We consider an autonomous system which undergoes to an AH bifurcation pattern, and we assume that the coefficients of its linear part are subject to a rapidly varying non-autonomous perturbation. key observation is that, even if the functions Hi have 0 average, using Jensen’s inequality we
see that
¯
a= lim
τ→+∞
1 τ
Z τ
0 a(σ)dσ =1+Aµ¯ , b¯ = lim
τ→+∞
1 τ
Z τ
0 b(σ)dσ =1+Bµ¯ (A.4) where ¯A>0, ¯B>0.
Therefore, if we consider the averaged system where we replacea,bby ¯a, ¯band we pass to polar coordinates,we observe an increment in the rate of rotation of orderµ(A¯+B¯)/2even if the perturbationhi(t)of the coefficients have0average. Namely if we set ¯r = [ay¯ 21+by¯ 22]1/2, and ¯θ =arctan
√by¯ 2
√ay¯ 1
we get dr¯ dt =µ
n
e¯r−r¯3(1+µC1(θ¯)) +r¯5C2(θ,¯ e)o, dθ¯
dt =µ pa¯b¯n
1+ω(e)r¯2C3(r, ¯¯ θ) +r¯4C4(r, ¯¯ θ,e)o
(A.5)
where the functionsCi are uniformly bounded in their respective variables.
From this computation we also see that the attracting invariant manifold, if exists, “tend to assume a more elliptic like shape”, i.e. it is a smallτdependent deformation of the ellipse
¯
ay21+by¯ 22 =e, that is ¯r =√ e.
Following [2] we point out that if we replace the functionshi(t
µ)in (A.1) by the large and fast varying functions µαhi(t/µ)(whereαis a constant) we obtain that ¯a=1+αA, ¯¯ b=1+αB¯ in (A.4), i.e. we have a macroscopic change in the speed of rotation close to the origin (of orderO(1)). However, ifαis not small enough, the whole structure of the bifurcation pattern might be washed away when the whole non-autonomous perturbation problem is considered (and not just its averaged system as in [1,2]): in fact we needα= o(e)to apply our techniques and to obtain the results in Sections3.1, 3.2. This might depend on the method of proof we used, but we believe that some smallness condition onαis most probably needed.
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